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Mirrors > Home > MPE Home > Th. List > opprval | Structured version Visualization version GIF version |
Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
opprval.1 | ⊢ 𝐵 = (Base‘𝑅) |
opprval.2 | ⊢ · = (.r‘𝑅) |
opprval.3 | ⊢ 𝑂 = (oppr‘𝑅) |
Ref | Expression |
---|---|
opprval | ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprval.3 | . 2 ⊢ 𝑂 = (oppr‘𝑅) | |
2 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅) | |
3 | fveq2 6663 | . . . . . . . 8 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = (.r‘𝑅)) | |
4 | opprval.2 | . . . . . . . 8 ⊢ · = (.r‘𝑅) | |
5 | 3, 4 | syl6eqr 2871 | . . . . . . 7 ⊢ (𝑥 = 𝑅 → (.r‘𝑥) = · ) |
6 | 5 | tposeqd 7884 | . . . . . 6 ⊢ (𝑥 = 𝑅 → tpos (.r‘𝑥) = tpos · ) |
7 | 6 | opeq2d 4802 | . . . . 5 ⊢ (𝑥 = 𝑅 → 〈(.r‘ndx), tpos (.r‘𝑥)〉 = 〈(.r‘ndx), tpos · 〉) |
8 | 2, 7 | oveq12d 7163 | . . . 4 ⊢ (𝑥 = 𝑅 → (𝑥 sSet 〈(.r‘ndx), tpos (.r‘𝑥)〉) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
9 | df-oppr 19302 | . . . 4 ⊢ oppr = (𝑥 ∈ V ↦ (𝑥 sSet 〈(.r‘ndx), tpos (.r‘𝑥)〉)) | |
10 | ovex 7178 | . . . 4 ⊢ (𝑅 sSet 〈(.r‘ndx), tpos · 〉) ∈ V | |
11 | 8, 9, 10 | fvmpt 6761 | . . 3 ⊢ (𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
12 | fvprc 6656 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = ∅) | |
13 | reldmsets 16499 | . . . . 5 ⊢ Rel dom sSet | |
14 | 13 | ovprc1 7184 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (𝑅 sSet 〈(.r‘ndx), tpos · 〉) = ∅) |
15 | 12, 14 | eqtr4d 2856 | . . 3 ⊢ (¬ 𝑅 ∈ V → (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉)) |
16 | 11, 15 | pm2.61i 183 | . 2 ⊢ (oppr‘𝑅) = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
17 | 1, 16 | eqtri 2841 | 1 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos · 〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 〈cop 4563 ‘cfv 6348 (class class class)co 7145 tpos ctpos 7880 ndxcnx 16468 sSet csts 16469 Basecbs 16471 .rcmulr 16554 opprcoppr 19301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-tpos 7881 df-sets 16478 df-oppr 19302 |
This theorem is referenced by: opprmulfval 19304 opprlem 19307 |
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